I've noticed that enemies stop re-spawning after a while, and searchning oline suggest this will happen after an enemy has re-spawned 15 times. I'm currently not in a good spot to test this myself in-game though. I'm also interested if there is anyway to make enemies start re-spawning again after they've reached their re-spawn limit. It would seem there is some item which can cause enemies to re-spawn (and supposedly make them stronger), but I can't find out any details on what this item is called.

What I'm looking for then is confirmation on the re-spawn limit, the name of this item which can reset the re-spawn limit, which enemies the item will cause to re-spawn, and if there are any additional effects the item has outside of just re-spawning enemies.

2 Answers
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Some of this information will overlap with the previous answer, but as that one doesn't address your question in full and relies mostly on speculation, I'll try to fill those gaps.

Confirmation on the respawn limit: There are lots of people out there currently debating the relative merits and demerits of the system, so I think the existence of the limit is pretty well established. So far, the consensus seems to be that the limit is approximately 15 deaths/spawns, but this won't be known for certain until the PC version comes out and can be hacked, or until someone actually sits down, does a playthrough, and records all of the data.

Name of the item which can reset the respawn limit: As @kiilla mentions, this is the Bonfire Ascetic.

Which enemies the item will cause to respawn: It will respawn all enemies in the bonfire's area, including bosses. (More accurately, it resets the respawn limit on all enemies in the area -- meaning that if you've killed one enemy seven out of fifteen times, it will once again have fifteen lives as well).

Additional effects the item has outside of just respawning enemies:This thread explains it much better and more succinctly than I could. The most important thing to understand is that it is actually making the area one difficulty level higher than it's currently set to (e.g., if you're playing on NG, the area will be set to NG+). Relevant portion of the thread quoted below.

Loot on dead bodies is respawned (but not loot in chests, those stay opened and empty)

Doors stay open, and any keys you've collected remain effective

If there are any NG+ enemy types in the area, they will spawn

NG+ BP enemies and NG+ BP NPCs will spawn in the area (the BP NPCs will not respawn if killed unless another ascetic is burned)

Enemies have more hp, deal more damage, and drop more souls on death

If there is a boss within the bonfire's area, it will respawn

If the boss drops any additional loot in NG+ or higher, it will drop that loot when killed in addition to the loot it normally drops (boss souls, etc)

Bosses have more hp, deal more damage, and drop more souls on death

Note that the effects of the Bonfire Ascetic are permanent and will continue into future difficulties as well. This means that if you use one in NG (thus making an area into NG+ difficulty), the area will be in NG++ difficulty when you move to NG+.